Smooth weighted zero-sum constants

Abstract: Let $A\subseteq\mathbb Z_n$ be a weight-set and $S=(x_1,x_2,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is a smooth $A$-weighted zero-sum sequence if there exists $(a_1,\ldots,a_k)\in A^k$ such that we have $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. It is easy to see that if $S$ is a smooth $A$-weighted zero-sum sequence, then for every $y\in \mathbb Z_n$ the sequence $S+y=(x_1+y,\ldots,x_k+y)$ is also a smooth $A$-weighted zero-sum sequence. From the well known EGZ-theorem it follows that if $S$ has length at least $2n-1$, then $S$ has a smooth $A$-weighted zero-sum subsequence of length $n$. The constant $\bar E_A$ is defined to be the smallest positive integer $k$ such that any sequence of length $k$ in $\mathbb Z_n$ has a smooth $A$-weighted zero-sum subsequence of length $n$. A sequence in $\mathbb Z_n$ of length $\bar E_A-1$ which does not have any smooth $A$-weighted zero-sum subsequence of length $n$ is called an $\bar E$-extremal sequence for $A$. For every $n$ we consider the weight-sets ${1}$ and $\mathbb Z_n\setminus{0}$. When $n$ is an odd prime $p$ we consider the weight-set $Q_p$ of all non-zero quadratic residues. We also study the related constants $\bar C_A$ and $\bar D_A$.